Talbot effect for the periodic Benjamin--Ono equation

Abstract

We study the Talbot effect for the periodic Benjamin--Ono equation with rough initial data. Using the smoothing theorem of G\'erard--Kappeler--Topalov for Tao's gauge transform, we reduce the singularity analysis to an explicit quadratic gauge profile. For general bounded-variation data we obtain a rational-time gauge-Hardy representation. For a natural subclass of Talbot-admissible data, whose initial gauge profile has only finitely many edge singularities, this representation becomes a finite sum of logarithmic kernels and one-jump kernels, up to a continuous remainder. We show that finite-jump, piecewise smooth BV data are Talbot-admissible; in particular, this applies to the square wave. At irrational times, we prove continuity under a finite Diophantine type condition, and we also exhibit a Liouville obstruction showing that the corresponding one-sided gauge profile need not be continuous at all irrational times. The results give a rigorous structural result that is qualitatively consistent with the numerical profiles of Alama Bronsard--Laurens.